{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Dr. Katz DEq Homework Solutions 95

# Dr. Katz DEq Homework Solutions 95 - Algebraic Solutions of...

This preview shows page 1. Sign up to view the full content.

Algebraic Solutions of Differential Equations 95 To conclude, we simply observe that as c-ar c-br the inequalities in (6.6.0.15) and (6.6.0.16), already established, are neces- sarily strict. This concludes the proof of (6.6.0). (6.6.1) Combining 6.3 and 6.6.0, we are "reduced" to proving the implication (6.2.6)~ (6.2.4) under the additional hypothesis that the rational numbers a, b, c satisfy (6.6.1.0) aCZ, b6Z, c~Z, c-aCZ, c-bc~Z, a-b6Z, c-a-b~Z. (6.6.2) Corollary. In order that rational numbers a, b, c satisfy (6.2.6) and (6.6.1.0), it is necessary and sufficient that for any A eZ which is invertible modulo N for a common denominator N of a, b, c, the rational numbers A a, A b, A c satisfy (6.2.6) and (6.6.1.0). Proof The sufficiency is clear; take A = 1. For the necessity, (6.6.1.0) will still hold because d is invertible mod N, and the condition (6.6.0.2) is obviously invariant under (a, b, c)--* (A a, A b, A c) for such A. By (6.6.0), (6.6.0.2) and (6.6.1.0) imply (6.2.6). (6.6.3) Corollary. In order that rational numbers a, b, c satisfy (6.2.6) and (6.6.1.0), it is necessary and sufficient that for any integers r, s, t, the
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}